George Berkeley is famous for his attack on Newton’s mathematics. In the short essay entitled The analyst Berkeley criticises the centrepiece of Newton’s new mathematical understanding of the universe, the calculus. In particular Berkeley argues against the legitimacy of the concept of the infinitesimal. Berkeley claims that these infinitesimals are a metaphysical and logical nonsense1 and that any mathematics that relies upon treating infinitesimals as real is wrong. These infinitesimals are nothing more than the “ghosts of departed quantities”2 and as such are no more real than ghosts in general. The gist of the problem Berkeley raises is roughly as follows – if the calculus offers an explanation for how the Universe works, and yet it relies upon these ghostly entities called infinitesimals, then it is no more and no less metaphysical than an alternative explanation that relies upon an alternative ghostly entity, perhaps God for example. The fundamental aim of this critique is to undermine any anti-religious implications that might be drawn from Newtonian mathematics. If the Newtonians thought that their new method offered a non-metaphysical and scientific route to the truth about the Universe, Berkeley wants to claim, then they are simply wrong. The idea is that implicit in Newtonian science is a set of commitments to features of reality (infinitesimals) that is just as metaphysical as religions commitment to its own, particular, features of reality (supernatural entities such as God, angels, demons or spirit and soul).
Berkeley is attacking what he takes to be the existential commitments involved in calculus. The route by which he does this is through charging the method with a duplicitous existential commitment. At one moment, he argues, the calculations define a variable as nothing and then at another point the same variable is defined as something. This logical duplicity covers up an existential impossibility, namely that something cannot be both something and nothing. The argument of The analyst illustrates this process quite clearly. In the first 8 sections the argument rests on the claim that infinitesimals are incapable of clear and distinct conception. From section 9, however, we find that the argument rests no longer on the conception of the infinitesimal but instead turns to the operation and definition of the rules of the calculations. The final move, grounded on the claim that “whether you argue in symbols or in words, the rules of right reason are the same” (Section XV), brings together the detailed analysis of the way the rules work with the problem of meaning. “Nothing is easier than to assign names, signs or expressions to these fluxions, and it is not difficult to compute and operate by means of such signs. But it will be found much more difficult, to omit the signs and yet retain in our minds the things, which we suppose to be signified by them” (Section XXXVII). Berkeley’s argument is in effect that the original formulations of the arguments at the heart of the calculus could not be correct because even if they could be codified into an operational set of rules they rest upon a foundation of contradictory meanings.
This argument provokes a reaction within mathematics which is highly productive, part of the historical background to the development of numerous mathematical innovations during the nineteenth century which eventually result in a concept of calculus that is ‘purged’ of the infinitesimal. This purge attempts to remove latent existential commitments by replacing any formulation which contains such commitments with procedures. Clear rules enable a concept to be defined not in terms of meaning but in terms of how it works. The general history of calculus understands this ongoing process as one of moving from the mysticism (of meanings) to the rigor (of rules). In the process ‘calculus’ becomes renamed ‘analysis’.
The shift from ‘meanings’ to ‘rigor’ involves an expunging of existential commitments in favour of a method of proving. A rigorous proof is one where the rules can be checked and followed and confirmation involves checking that the rules have been followed, nothing more. The concept of rigor is algorithmic in essence, enabling a step-by-step check of clear and distinct steps to be undertaken. It results, in modern mathematics, in proof-checking being carried out by mechanical computers who are capable of algorithmic calculation at a far greater rate and degree of accuracy than any human.
This brief story now needs to be complicated. Unshakeable proofs enable epistemological authority. Yet clear explanations that make sense of observations also attract authority and in the case of Newton it is the latter that initially holds sway. The gravitational theories of Newton developed in the Principia are dependent on the calculus. Their authority rests, however, first and foremost on the observational testing that was enabled. Newton tells the story of the cosmos in a mechanical method that can be seen to be true. The authority of Newton rests upon the observation of the correctness of his mathematical calculation whilst yet hand Berkeley’s challenge undermines the foundations of these very same calculations. For a long time the problem of foundations is pushed aside and does not return for a century or so. The manner in which it returns, however, is illuminating. In 1784 the ‘prize problem’ of the Berlin Academy is the question of the foundations of the calculus. Judith Grabiner tells us some of the background of this story, involving the way in which the problem of foundations had developed gradually during the eighteenth century3. The posing of the Berlin Academy prize problem, in the same year that Kant responded to the philosophical problem of the Enlightenment, arises at a time in which the problem of foundations is not simply theoretical but has begun to become one of authority, specifically, of the authority of the teacher.
Grabiner argues that in the late eighteenth century there is a shift in the social position in which mathematics is taught. From being a feature of the declining Royal Courts mathematics moves into the public realm, in places such as the Ecole Polytechnique, where courses on calculus were being taught to engineers. Teaching, claims Grabiner, “forces one’s attention to basic questions”4. The reasons for this might be thought of in two different ways. Teaching forces the need to explain ideas simply and from the beginning. In order to explain to a non-initiate the arguments and ideas of a subject one needs to return to that which has often been taken for granted by the teacher as they moved through their own educational development. However this ‘force’ is not simply one of a pure motive of clarity but is deeply embroiled in the need for the teacher to establish an authority in the face of questioning. If the subject being taught is one that supposedly rests on nothing but rational clarity – as is the case with mathematics – then that rational clarity better be there for all to see. There is nothing worse, in this situation, that having to rely upon a vague and curiously self-contradictory foundation. The problem of foundations becomes a problem of authority, a problem of the teacher being able to explain clearly and accurately exactly what they mean. For this reason, as Grabiner suggests, it is no coincidence that a whole series of mathematical developments occur as a response to the practical and qualitative relationship a practitioner has with students. In the case of the calculus the rigorous development of the concept of limits arises when Augustin-Louis Cauchy teaches at the Ecole Polytechnique. In 1821 Cauchy produces the Cours d’analyse, the book that is now taken to be the first real step to the modern rigorous calculus.
The point of this story about the origins of the calculus, a subject that is replete with fascinating and complex problems regarding reality, is to suggest that reason is deeply embedded in the practice of real living. The basic claim is that to understand the course of rational knowledge it is useless to simply look at the internal consistency of that knowledge. Reason is not some holistic self-contained set of inferential relations that are gradually cleaned and polished as we proceed. Reason is an engaged, embedded, process. It is engaged and embedded in the practices of real living which involve social dynamics, individual affects and co-ordinations between these social dynamics, individual affects and material necessities. Berkeley challenges the internal consistency of a theoretical construct. This challenge is not accepted and its refusal begins to produce various unsatisfactory responses, unsatisfactory in the sense that they can not dismiss Berkeley’s problem. Yet Berkeley’s problem does not undermine the social dynamics that rest upon the observational efficacy of the Newtonian model. The model continues in spite of such rational challenges to produce a rational challenge to religious understandings. Only gradually, as the transmission of that rational challenge seeps its way into the practices of engineering classes, does the rational challenge to theoretical foundations become a problem that needs solving. In understanding our reason what we must be cautious of is not simple error or mistaken inference but the forgetting of the bodies of reasonable beings. The real ghosts of departed quantities are not the somethings taken for nothings that Berkeley points to but the actual bodies of the reasonable beings involved in the actual processes of knowledge.
1See Blaszczyk, Katz and Sherry, Ten misconceptions from the history of analysis and their debunking, arXiv:1202.4153v1
2Berkeley, The analyst, Section XXXV.
3Judith Grabiner, Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus, in The American Mathematical Monthly, Vol.90, number 3, March 1983, pp185-194